Math Problem Statement
Solve the system of equations:
y = (1/2)x - 3 and x - 2y = 6.
Solution
The system of equations provided in the image is:
y = \frac{1}{2}x - 3 \\ x - 2y = 6 \end{cases}$$ To solve this system, we can use the **substitution method**. ### Step 1: Substitute $$y = \frac{1}{2}x - 3$$ into the second equation. The second equation is: $$x - 2y = 6$$ Substitute $$y = \frac{1}{2}x - 3$$ into this: $$x - 2\left(\frac{1}{2}x - 3\right) = 6$$ Simplify the expression inside the parentheses: $$x - \left(x - 6\right) = 6$$ $$x - x + 6 = 6$$ This simplifies to: $$6 = 6$$ ### Step 2: Interpretation of the result Since the equation reduces to $$6 = 6$$, which is a **true statement** but does not involve any variables, this indicates that the system of equations has **infinitely many solutions**. This happens when both equations represent the same line. Therefore, the system has **infinitely many solutions**, meaning the two lines coincide. ### Graphical Interpretation: If we were to graph these two equations, they would lie on the same line, reinforcing that there are infinitely many points where the lines intersect. --- Would you like me to go into more detail or clarify any step further? Here are 5 follow-up questions for you: 1. How can you identify if a system has infinitely many solutions or no solution just from its equations? 2. What does it mean graphically when two lines coincide? 3. Can you think of an example of a system of equations that has no solution? 4. How would you solve this system using the elimination method? 5. What are the implications of the result "6 = 6" in terms of systems of equations? **Tip:** When solving systems of linear equations, always look for ways to simplify the system early on, such as substitution or elimination methods, which can make the process faster.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Substitution Method
Formulas
Slope-intercept form y = mx + b
System of equations: Solve using substitution or elimination
Theorems
Infinitely many solutions theorem for dependent linear systems
Suitable Grade Level
Grades 9-11